Suppose that we have a real phenomenon. The phenomenon produces events (synonym: outcomes ).

Transcription

1 p. 5/44 Modeling Suppose that we have a real phenomenon. The phenomenon produces events (synonym: outcomes ). Phenomenon Event, outcome We view a (deterministic) model for the phenomenon as a prescription of which events can occur, and which events cannot occur.

2 p. 8/44 The behavior A mathematical model : a pair (U,B) with U the universum of events B U the behavior of the model allowed B U possible forbidden

3 p. 10/44 Discrete event phenomena If U is a finite set, or strings of elements from a finite set, we speak about discrete event systems (DESs). Phenomenon Examples: Words in a natural language Sentences in a natural language DNA sequences LATEX code Block codes

4 p. 11/44 Discrete event phenomena Words in a natural language U = A (:= all finite strings with letters from A) with A = {a,...,z,a,...,z}. B = all words recognized by the spelling checker, for example, behavior B, SPQR / B. B is basically specified by enumeration. Sentences in a natural language U = A (:= all finite strings with letters from A) with A = {a,...,z,a,...,z,,.;: ()!?, etc.}. B = all legal sentences. Specifying B is a complicated matter, involving grammars.

5 p. 13/44 Continuous phenomena If U is a (subset of a) finite-dimensional real (or complex) vector space, we speak about continuous models. Phenomenon Examples: The gas law A spring The gravitational attraction of two bodies A resistor

6 p. 13/44 Continuous phenomena A resistor Event: voltage V, current I. Throughout, we take the current positive when it runs into the circuit, and we take the voltage positive when it goes from higher to lower potential. I + R V Georg Ohm ( ) U = R R; B = {(V,I) R R V = RI } (Ohm s law).

7 p. 14/44 Dynamical phenomena If U is a set of functions of time, we speak about dynamical models. Phenomenon signal space Examples: Inductors, capacitors Kepler s laws Newton s second law Convolutional codes time

8 Dynamical phenomena Inductors and capacitors Event: voltage and current as a function of time. I + I + L V C V U = (R R) R ; B = {(V,I) : R R R L d dt I = V } (inductor), B = {(V,I) : R R R C d dt V = I } (capacitor). p. 14/44

9 p. 14/44 Kepler s laws Dynamical phenomena Event: the position of a planet as a function of time. 1 year PLANET SUN 7 months K1: ellipse, sun in focus, K2: equal areas in equal times, K3: square of the period = third power of major axis. U = (R 3 ) R ; B = { q : R R 3 K1, K2, & K3 hold }. Johannes Kepler ( )

10 Dynamical phenomena Newton s second law Event: the position of a pointmass and the force acting on it, both as a function of time. F M q U = (R 3 R 3 ) R ; B = {( q, F) : R R 3 R 3 F = M d2 dt 2 q }. Newton painted by William Blake p. 14/44

11 p. 16/44 Distributed phenomena If U is a set of functions of space and time, we speak about distributed parameter systems. Phenomenon signal space Examples: Heat diffusion Wave equation Maxwell s equations space time

12 Distributed phenomena Maxwell s equations Event: electric field, magnetic field, current density, charge density as a function of time and space. E = 1 ε 0 ρ, E = t B, B = 0, James Clerk Maxwell ( ) c 2 B = 1 ε 0 j + t E. U = (R 3 R 3 R 3 R) R4 ; B = {( E, B, j,ρ) : R R 3 R 3 R 3 R 3 R Maxwell s equations are satisfied }. p. 17/44

13 p. 19/44 Behavioral models The behavior captures the essence of what a model is. The behavior is all there is. Equivalence of models, properties of models, symmetries, system identification, etc. must all refer to the behavior. Every good scientific theory is prohibition: it forbids certain things to happen. The more it forbids, the better it is. Replace scientific theory by mathematical model. Karl Popper ( )

14 p. 21/44 The dynamic behavior In dynamical systems, the events are maps, with the time-axis as domain. The events are functions of time. Phenomenon signal space time It is convenient to distinguish, in the notation, the domain of the event maps, the time set, and the codomain, the signal space, that is, the set where the functions take on their values.

15 p. 22/44 The dynamic behavior Definition: A dynamical system : (T, W, B), with T R the time set, W the signal space, B W T the behavior, that is, B is a family of maps from T to W. w : T W B means: the model allows the trajectory w, w : T W / B means: the model forbids the trajectory w.

16 p. 22/44 The dynamic behavior Definition: A dynamical system : (T, W, B), with T R the time set, W the signal space, B W T the behavior, that is, B is a family of maps from T to W. w : T W B means: the model allows the trajectory w, w : T W / B means: the model forbids the trajectory w. Mostly, T = R,R + := [0, ),Z, or N := {0,1,2,...}, W = (a subset of) R w, for some w N, B is then a family of trajectories taking values in a finite-dimensional real vector space. T = R or R + continuous-time systems, T = Z or N discrete-time systems.

17 p. 23/44 Dynamical systems described by differential equations Consider the ODE f w, with d2 dn w, w,..., ddt dt2 dt n w = 0, ( ) f : W R w R w R w n times Some may prefer to write f w, d2 dn w, w,..., ddt dt2 dt n w R, W R w. = 0, instead of ( ), but we leave the notation to puritans.

18 p. 26/44 Linearity and time-invariance The dynamical system Σ = (T,W,B) is said to be linear : W is a vector space (over a field F) and [[w 1,w 2 B and α F]] [[w 1 + αw 2 B]]. Linearity the superposition principle holds.

19 p. 26/44 Linearity and time-invariance The dynamical system Σ = (T,W,B) is said to be time-invariant : T = R,R +,Z, or N, and [[w B and t T]] [[σ t w B]]. σ t denotes the backwards t-shift, defined as σ t w : T W, σ t w(t ) := w(t +t). σ t w w t time Time-invariance shifts of legal trajectories are legal.

20 p. 27/44 Autonomous systems The dynamical system Σ = (T,W,B), with T = R or Z, is said to be autonomous : [[w 1,w 2 B, and w 1 (t) = w 2 (t) for t < 0]] [[w 1 = w 2 ]].

21 p. 28/44 Autonomous in a picture W time PAST W time FUTURE autonomous : the past implies the future.

22 p. 29/44 Stability The dynamical system Σ = (T, W, B), with T = R,[0, ), Z, or N, and W a normed vector space (for simplicity), is said to be stable : [[w B]] [[w(t) 0 for t ]].

23 Stability The dynamical system Σ = (T, W, B), with T = R,[0, ), Z, or N, and W a normed vector space (for simplicity), is said to be stable : [[w B]] [[w(t) 0 for t ]]. In a picture W time stability : all trajectories go to 0. Sometimes this is referred to as asymptotic stability. There exist numerous other stability concepts for dynamical systems! p. 29/44

24 p. 30/44 Controllability The time-invariant (to avoid irrelevant complications) dynamical system Σ = (T,W,B), with T = R or Z, is said to be controllable : for all w 1,w 2 B, there exist T T,T 0, and w B, such that w(t) = w 1 (t) for t < 0; w 2 (t T) for t T.

25 p. 31/44 Controllability in a picture W w 1,w 2 B w 1 0 w 2 time

26 p. 31/44 Controllability in a picture W w 1,w 2 B w 1 0 w 2 time W transition W w B w 1 w 0 w T time σ T w 2 w controllability : concatenability of trajectories after a delay

27 p. 32/44 Stabilizability The dynamical system Σ = (T,W,B), with T = R or Z, and W a normed vector space (for simplicity), is said to be stabilizable : for all w B, there exist w B, such that w (t) = w(t) for t < 0, w (t) 0 for t.

28 p. 33/44 Stabilizability in a picture w w W 0 w time stabilizability : all trajectories can be steered to 0.

29 p. 34/44 Observability observed w 1 System w 2 to-be-deduced Consider the dynamical system Σ = (T,W 1 W 2,B). w 2 is said to be observable from w 1 in Σ : [[(w 1,w 2 ),(w 1,w 2) B and w 1 = w 1]] [[w 2 = w 2]].

30 p. 34/44 Observability observed w 1 System w 2 to-be-deduced Consider the dynamical system Σ = (T,W 1 W 2,B). w 2 is said to be observable from w 1 in Σ : [[(w 1,w 2 ),(w 1,w 2) B and w 1 = w 1]] [[w 2 = w 2]]. Observability : w 2 may be deduced from w 1.!!! Knowing the laws of the system!!!

31 p. 35/44 Observability in a picture W 1 F w 1 time W 2 w 2 time Equivalently, there exists a map F : W T 1 WT 2, such that [[(w 1,w 2 ) B]] [[w 2 = F(w 1 )]].

32 p. 36/44 Detectability Consider the dynamical system Σ = (T,W 1 W 2,B), with T = R,R +,Z, or N, and W a normed vector space (for simplicity). w 2 is said to be detectable from w 1 in Σ : [[(w 1,w 2 ),(w 1,w 2 ) B and w 1 = w 1 ]] [[w 2 (t) w 2 (t) 0 for t ]]. Detectability : w 2 can be asymptotically deduced from w 1.

33 p. 39/44 State equations We now discuss how state models fit in. d x = f (x,u), y = h(x,u), w = dt u, ( ) y with u : R U the input, y : R Y the output, and x : R X the state. In particular, the linear case, these systems are parametrized by the 4 matrices (A,B,C,D) d u x = Ax + Bu, y = Cx + Du, w =, dt y with A R n n,b R n m,c R p n,d R p m. These models have dominated linear system theory since the 1960 s.

34 p. 39/44 State equations We now discuss how state models fit in. d x = f (x,u), y = h(x,u), w = dt u, ( ) y with u : R U the input, y : R Y the output, and x : R X the state. It is common to view state space systems as models to describe the input/output behavior by means of input/state/output equations, with the state as latent variable. Define B extended := {(u,y,x) : R U Y X ( ) holds}, B := {(u,y) : R U Y x : R X such that ( ) holds}.

35 p. 40/44 State controllability State models propagated in the 1960 s under the influence of R.E. Kalman. Especially important in this development were the notions of state controllability and state observability. Rudolf Kalman (1930 )

36 p. 40/44 State controllability ( ) is said to be state controllable if for all x 1,x 2 X, there exists T 0, u : R U, and x : R X, such that 1. d dt x(t) = f (x(t),u(t)) for 0 t T, 2. x(0) = x 1, 3. x(t) = x 2. X x 2 x 1

37 State controllability ( ) is said to be state controllable if for all x 1,x 2 X, there exists T 0, u : R U, and x : R X, such that 1. d dt x(t) = f (x(t),u(t)) for 0 t T, 2. x(0) = x 1, 3. x(t) = x 2. It is easy to prove that [[state controllability]] [[behavioral controllability of B extended ]]. [[state controllability]] [[behavioral controllability of B]]. Behavioral controllability makes controllability into a genuine, an intrinsic, representation independent system property. p. 40/44

38 p. 41/44 State observability ( ) is said to be state observable if [[(u,y,x 1 ),(u,y,x 2 ) B extended ]] [[x 1 (0) = x 2 (0)]].

39 p. 41/44 State observability ( ) is said to be state observable if [[(u,y,x 1 ),(u,y,x 2 ) B extended ]] [[x 1 (0) = x 2 (0)]]. It is easy to prove that [[state observability]] [[behavioral observability of B extended ]], with (u,y) as observed variables, and x as to-be-deduced variable.

40 p. 41/44 State observability ( ) is said to be state observable if [[(u,y,x 1 ),(u,y,x 2 ) B extended ]] [[x 1 (0) = x 2 (0)]]. It is easy to prove that [[state observability]] [[behavioral observability of B extended ]], with (u,y) as observed variables, and x as to-be-deduced variable. Behavioral controllability and observability are meaningful generalizations of state controllability and observability. Why should we be so interested in the state?

41 p. 43/44 Summary A phenomenon produces events, outcomes. the universum of events U. A mathematical model specifies a subset B of U. B is the behavior and specifies which events can occur, according to the model. In dynamical systems, the events are maps from the time set to the signal space. Controllability, observability, and similar properties can be nicely defined within this setting. State models are a more structured class of dynamical systems.